Stress Transfer From the Matrix to the Fibre in a Fragmentation Test

16 Pages·1999·0.3 MB·English

by  John Nairn

Stress Transfer From the Matrix to the
Fibre in a Fragmentation Test: Raman
Experiments and Analytical Modeling
ALKIS PAIPETIS AND COSTAS GALIOTIS
University of London
Queen Mary and Westfleld College
Mile End Road
London E1 4NS, UK
YUNG CHING LIU AND JOHN A. NAIRN⁄
Material Science and Engineering
University of Utah
Salt Lake City, Utah 84112, USA
(Received July 8, 1997)
(Revised March 9, 1998)
ABSTRACT: The stress transfer properties of the flbre/matrix interface in the single flbre fragmentation test
were investigated. Two carbon flbre-resin systems involving epoxy-sized and unsized flbres, were examined. Axial
flbrestressdataatresolutionsoftheorderofonemicron,wereobtainedwiththetechniqueofRemoteLaserRaman
Microscopy. Subsequent analytical modeling of the data was performed using a Bessel-Fourier Series stress analysis
approach. The analysis provides a nearly exact solution for the stress fleld in the fragmentation test and simultane-
ously accounts for damaged or imperfect interfaces through the use of an interface parameter D . All data were flt
s
usingatwozonemodelinordertoaccountforthepropagationofinterfacialdamageasafunctionofappliedstrain.
The fltting process was used to determine D(i), the interface parameter in undamaged zones, D(d), the interface
s s
parameterinthedamagezonesnearflbrebreaks,andl ,thelengthofthedamagezones,allasafunctionofapplied
d
strain. The interface parameter in the undamaged zones, D(i), was independent of applied strain. We claim D(i) is
s s
a good property for comparing flbre/matrix interfaces and the most relevant property for predicting the role to the
interface in real laminates.
KEY WORDS: Raman Microscopy, Stress Transfer, Interface, Imperfect Interface
INTRODUCTION
I
N THE SINGLE-FIBRE fragmentation test, a single flbre is embedded in a large amount of matrix and
the specimen is loaded in tension. The tensile loading causes the flbre to break into fragments; the test
is continued until the fragmentation process ceases [1]. The fragmentation process is in(cid:176)uenced by the
properties of the flbre/matrix interface; thus the single-flbre fragmentation test is widely used as a method
for assessing interfacial properties. The question remains: how does one extract quantitative interfacial
property information from the results of single-flbre fragmentation tests? Previous investigators have relied
solely on fragment length vs. strain data or sometimes only on critical fragment length data which is the
fragment length when the fragmentation process ceases. Such data is interpreted using simplistic elastic
⁄Authortowhomcorrespondenceshouldbeaddressed.
Journal of Composite Materials, Vol. 33, No. 4, 377{399/1999 377
Author Prepared Reprint
(cid:176)c 1998 Technomic Publishing Co., Inc.
378 Alkis Paipetis, Costas Galiotis, Yung Ching Liu, and John A. Nairn
plastic models [2], or elastic-plastic models in combination with simplistic, elastic, shear-lag models [3]. It
is unlikely that such models can extract useful information from fragmentation data. By \useful," we mean
quantitative interfacial properties that can be used to predict the role of the interface in real laminates. We
claim it is additionally unlikely that any model could extract unambiguous information from fragmentation
data alone. We need improved models and we need additional experimental observations on fragmentation
specimens.
Here we have used Laser Raman Spectroscopy (LRS) during a fragmentation test as a tool to provide
supplemental experimental observations about the fragmentation process. As has been demonstrated, LRS
allows us to measure axial flbre stress in a composite with a spatial resolution of the order of 1 „m [4].
The fragmentation specimen provides an ideal conflguration for LRS studies for both long [5, 6] and short
reinforcing flbres [7]. Thus we can measure the axial flbre stress in fragments at any strain level during the
fragmentationprocess. PreviousLRSofembedded,singleflbreshasshownthatthesimplisticassumptionsof
elasticplasticmodelsareinaccuratebecausetheinterfacialshearstressisnotconstant[5,6]. LRSinregions
of elastic stress transfer show that shear-lag models empirically predict flbre/matrix stress transfer [5, 6, 7].
Recent theoretical results, however, have shown that shear-lag analysis breaks down for low flbre volume
fractions [8]. Thus, when shear-lag analysis is used to flt LRS results, the only information that results
is an empirical determination of the shear-lag parameter. This parameter, however, cannot be related to
interfacial properties for low flbre volume fraction specimens [8].
An improved stress analysis of the fragmentation test that can be useful in interpreting LRS results
has been derived using a Bessel-Fourier series stress function in a full, axisymmetric stress analysis [9{11].
In brief, the Bessel-Fourier analysis is an exact elasticity solution (i.e., the solution obeys equilibrium and
compatibility). It further obeys all boundary conditions exactly except one. The single approximation is
that the axial stress at the end of a broken flbre is equal to zero in an averaged sense instead of being
uniformly equal to zero. Most elasticity analyses of flbre/matrix stress transfer assume a perfect interface.
Such analyses are no help in interpreting fragmentation results because there is no interface property that
can be varied to predict the role of the interface. In contrast, the Bessel-Fourier analysis in Ref. [9] accounts
for an imperfect interface. The technique is to introduce an interface parameter, D , which is an efiective
s
stifiness for sliding of the flbre relative to the matrix under the in(cid:176)uence of interfacial shear stress [12]. A
value of D =1 corresponds to an perfect interface; a value ofD =0 corresponds to a debonded interface.
s s
Any other value of D represents an imperfect interface. We claim that using LRS to measure D during
s s
a fragmentation test is a potential method for characterizing the flbre/matrix interface [10]; the measured
interfacial parameters have the potential to be used for studying the role of the interface in real laminates.
In this study, fragmentation tests with simultaneous LRS experiments were run on sized and unsized
high modulus carbon flbres embedded in an epoxy resin. In a previous publication [4], the stress transfer
characteristics of the two systems have been studied and difierent mechanisms of interfacial failure have
been identifled. In brief, a zone of damage appears near flbre breaks. Within the damage zone the rate of
stress transfer is slower than when the interface is undamaged or in the central portion of fragments away
from the damage zones. Clearly no single interface parameter, D , can predict the observed change in stress
s
transfer properties between the damaged and undamaged zones. We thus modeled the experimental results
usingatwo-zone, Bessel-Fourieranalysis[9]. WithinthedamagezoneD =D(d) whereD(d) istheinterface
s s s
parameter for a damaged interface. We need not know precisely the mechanism of damage; we instead
regardameasuredvalueofD(d) asameasureoftheabilityofadamagedinterfacetotransferstressfromthe
s
matrixtotheflbre. SomephysicalefiectsthatmightplayaroleinthevalueofD(d) includeinterfacialcracks,
s
interfacialfriction, matrixyieldingneartheinterface, andmatrixcracks. Intheundamaged, centralportion
offragmentsD =D(i),whereD(i) istheinterfaceparameterforanundamagedorintactinterface. Wehave
s s s
measured D(d), D(i), and the size of the damage zone as a function of applied strain during a fragmentation
s s
test. D(d) decreases as strain increases indicating an increased impairment to stress transfer as interface
s
damage becomes more extensive. In contrast, D(i) is independent of applied strain. We claim that D(i) is
s s
preferredoverD(d) andovercriticalfragmentlengthexperimentsasameasureoftheflbre/matrixinterfacial
s
properties; it can be measured with LRS experiments.
Stress Transfer Fragmentation Test: Raman Experiments and Analytical Modeling 379
MATERIALS AND METHODS
Laser Raman Spectroscopy (LRS) may be used to measure stresses in flbres in composites due to the stress
or strain dependence of certain vibrational modes in the flbres [13]. Unique calibration curves are normally
produced by stretching individual carbon flbres and recording the position of the carbon flbre peak as a
function of either applied stress or strain. These calibration curves can afterwards be used to obtain stress
orstrainvaluesofothersimilarflbresembeddedincomposites. LRShasbeensuccessfullyusedinapplications
involving a wide range of materials and specimen geometries, from single-flbre model composites [14] to real
composite coupons [15]. LRS provides a unique tool for measuring internal stresses in composites.
Materials and Specimen Preparation
The sized and unsized high modulus flbres used in this study were provided by Soflcar. The M40B-
40B sized flbres are labeled MEBS; the M40B unsized flbres are labeled MUS. Both flbres have an efiective
diameterof6.6„m(asdeterminedbydensitymeasurements)andamodulusof390GPa. Theflbrestrengths
weredeterminedatfourgaugelengthsforeachflbretype. Bothflbreswerereportedtohaveastandardlevel
of oxidative treatment.
The flbres were embedded in an epoxy resin that was provided by Ciba-Geigy; it was a two part MY-
750/HY-951 epoxy system. The resin (MY-750: unmodifled liquid epoxy resin) and hardener (HY-951:
triethylene tetramine) were mixed at 40–C at a ratio 4:1, degassed for 10 min under full vacuum and poured
into silicon rubber dogbone moulds where individual flbres had previously been carefully aligned. The
single-flbre composites were cured for 2 hrs at 60–C, removed from the moulds and subsequently post-cured
at 120–C. The modulus, strength and approximate failure strain for the resin were 2.6 GPa, 65 MPa and
8%, respectively [4].
The cured dogbone samples were ground until the embedded flbre was approximately 100 „m away from
thesurfaceandthensubsequentlypolished. Astraingaugewasattachedtotheresinsurfaceandtheapplied
strain was derived from resistance measurements using a digital multimeter.
Specimen Testing and Raman Spectrum Acquisition
The LRS data acquisition was performed using the Remote Raman Microprobe (ReRaM). The ReRaM
hasbeendevelopedandtestedinhouse,anduses(cid:176)exiblewaveguidesforboththedeliveryandthecollection
of light [4]. In addition, micrometer stages allow for the translation of the ReRaM in all three axes down to
an accuracy of 1 „m. In this work, the laser was focused to a submicron spot on the specimen using a laser
power of approximately 1 mW. The collected Raman light was guided through an optical flbre to a SPEX
1000M single monochromator. The Raman signal was collected via a Wright Instrument CCD and stored
in a PC compatible computer. Spatial discrimination was achieved by magniflcation of the focal spot to flll
the optical flbre end face.
RamandatawerecollectedbyscanningtheRamanmicroprobepoint-by-pointalongtheembeddedflbre.
In general a section of the Fragmentation Gauge Length (FGL) of 2.5-3 mm in length, was scanned at
difierent increments of externally applied load. Within this \window", measurements were taken at steps of
1„minthevicinityofflbrebreakswhereflbrestressischangingrapidlyandthenatstepsof2,5and10„m
in regions away from flbre breaks where flbre stress changes more slowly.
The dogbone samples were mounted on a Hounsfleld universal testing machine, and strained to strain
levels of up to 5.00%. The strain rate was kept to a minimum (i.e., 0:25£10¡2 min¡1) in order to avoid
relaxation efiects. Fragmentation specimens with both sized and unsized specimens were tested until failure
which occurred at approximately 8% strain.
Bessel-Fourier Analysis of Experiments
LRS experiments on fragmentation specimens can measure the axial stress in the flbre as a function of
position along the flbre. To interpret these experiments, we flt the measured stress to a calculated stress
based on a three-dimensional, axisymmetric stress analysis of stress transfer in fragmentation specimens in
thepresenceofanimperfectinterface[9]. Thestresstransferanalysis,whichusesaBessel-Fourierseriesstress
function, accounts for an imperfect interface through an interface parameter, D . The flt to experimental
s
results allows us to measure D and therefore to get a direct measure of the stress-transfer properties of the
s
flbre/matrix interface.
380 Alkis Paipetis, Costas Galiotis, Yung Ching Liu, and John A. Nairn
The Bessel-Fourier series solution for the fragmentation specimen is described in detail in Ref. [9]; here
we only quote those parts of the solution required to interpret LRS experiments. The average axial flbre
stress in a fragment of length l is given by [9]:
‡ D E·
h(cid:190)zz;fi=ˆ1 1+ (cid:190)z(pz);f (1)
whereˆ1 istheaxialstressintheflbrebeforethereareanyflbrebreaks(thefar-fleldflbrestress)and(cid:190)(p)
zz;f
is the perturbation flbre stress. The far-fleld flbre stress is given by
‡ · ‡ ‡ · ·
2”A”m ¡ 1¡”T ¡ 1+”m EA(cid:190)0 + 2”A(fi ¡fi )+ 1¡”T + 1+”m (fi ¡fi ) E ¢T
ˆ1 = EA ET Em Em EA T m ET Em A m A (2)
2”A2 ¡ 1¡”T ¡ 1+”m
EA ET Em
whereE ,E ,” ,” ,fi ,andfi aretheaxialandtransversetensilemoduli,Poisson’sratios,andthermal
A T A T A T
expansion coe–cients of the flbre. The flbre is assumed to be transversely isotropic with the axial direction
along the axis of the flbre. The terms E , ” , and fi are the modulus, Poisson’s ratio, and thermal
m m m
expansion coe–cient of the assumed isotropic matrix. Finally, (cid:190) is the applied axial stress during the
0
fragmentation test and ¢T = T ¡T is the difierence between the specimen temperature, T , and the
s 0 s
stress-free temperature, T . The ¢T term accounts for residual stresses.
0
The solution in Eq. (1) is written as a superposition of the far-fleld stress with the perturbation stress.
Physically, (cid:190)(p) corresponds to the solution to the stress analysis problem of a single flbre fragment with
zz;f
a uniform compression stress of (cid:190) = ¡1 applied to the ends of the flbre at z = §l=2 (note: the origin of
zz
the z axis is taken to be at the middle of the flbre fragment). Superposition of the far-fleld stress with the
perturbation stress leads to solution in which the flbre stress is zero at flbre breaks. From Ref. [9], the total
flbre axial stress is
(cid:181) ¶
F(z)
h(cid:190)zz;fi=ˆ1 1¡ (3)
F(l=2)
where the function F(z) is deflned by
• (cid:181) ¶ (cid:181) ¶‚
E (1¡” ) ” a 1+” 1+” 1 1¡” 1+”
F(z) = ‡ A T · A m ¡ T ¡ T + m
2 1¡”T + 1+”m ¡ 2”A2 EA Em ET GT ET Em
ET Em EA• (cid:181) ¶ (cid:181) ¶ ‚
X1
d k z c I (fl ) c I (fl )
+ +2 cos i c ¡d 1 1i +c ¡d 1 2i (4)
2 r 1i s2 fl 2i s2 fl
i=1 f 1 1i 2 2i
and
a = ¡”A(1+”T) c = GEAA ¡”A(1+”T)
b = ”T1¡¡””AEA2EEAEATT ‡GEAA ¡”A· d = 2EGAT1(1¡¡”A”2ETEA)T
s21 = a+c+1p¡(2”adA2E+EATc)2¡4d s22 = a1+¡c¡”A2EpEAT(2ad+c)2¡4d (5)
ki = 2rfli… flji = skji
HereG andG aretheaxialandtransverseshearmodulioftheflbreandr istheradiusoftheflbre. The
A T f
function F(z) in Eq. (4) is written as a Bessel-Fourier series where I (x) is the modifled Bessel function of
1
theflrstkind[17]. Thetermsc andandc areconstantsoftheexpansionthatdependonflbreandmatrix
1i 2i
properties, on the fragment length, and on the flbre/matrix interface parameter (D ). All c and c terms
s 1i 2i
Stress Transfer Fragmentation Test: Raman Experiments and Analytical Modeling 381
may be found by solving the following matrix equation for each term in the Bessel-Fourier series:
0 ‡ · ‡ ·
a¡ 1 I1(fl1i) a¡ 1 I1(fl2i) ¡K (k )
BBB ‡a¡ 1 ·I (fls21)+s(11¡b)I1(fl1i) ‡a¡ 1 ·I (fls22)+s(21¡b)I1(fl2i) K (k )+1 Ki1(ki)
BBBB ‡s21 (b0s¡211)1i2IG1(Tflfl1i1)i·s21 fl1i ‡s22 (b0s¡221)2i2IG1(Tflfl2i2)i·s22 fl2i 0 ¡iK2G1m(kkii)ki
B@ ¡ s21G1A ¡‡d+E2A”Aa· I0(fl1i) ¡ s22G1A ¡‡d+E2A”Aa· I0(fl2i) K20G(kmi)
¡ 1 ¡a fl1iI1(fl1i) ¡ 1 ¡a fl2iI1(fl2i)
s21 Ds s22 Ds 0 1
10 1
2G1¡m2((¡11ki¡¡K2”1”m(mk)i)K)¡K¡1K0(20(k4Gk(i(km)i1i))¡¡+k”kimKiK)0K1((k0ki()ik)i)¢ CCABB@ cccc1234iiii CCA=BBBB@ (¡((¡¡111)i))8ii”4402A((EG11(kA¡+1Ti2+k””ki2TT”i2T))) CCCCA (6)
where G is the shear modulus of the matrix, K (x) and K (x) are modifled Bessel functions of the second
m 0 1
kind [17], and D is the imperfect interface parameter. The extra constants found when solving Eq. (6)
s
(c and c ) are needed to determine the matrix stresses [9]. They are not needed here when flnding only
3i 4i
the axial stress in the flbre. This solution is for an anisotropic flbre embedded in an isotropic matrix. The
solution does not include the solution for an isotropic flbre as a special case; the solution for an isotropic
flbre is given in Ref. [9]. All results presented here are for anisotropic carbon flbres and therefore only the
anisotropic-flbre result given above is needed. Finally, the solution is written as an inflnite series. This
seriesconvergesprovidedthenumberoftermsisequaltoorgreaterthantheaspectratio(l=2r )oftheflbre
f
fragment [9]. All analyses in this paper included su–cient terms to insure convergence.
Given flbre and matrix properties (see Table 1 for the properties used here for MEBS and MUS flbres
and for the MY750 epoxy matrix), Eq. (1) determines the axial stress in the flbre except for the unknown
interface parameter, D . The use of interface parameters in the analysis of composites was introduced by
s
Hashin[12]. Adeflnitionofaperfectinterfaceisthattheradialandaxialdisplacementsarecontinuousfrom
theflbretothematrix. Toincludeanimperfectinterface,thedisplacementsareallowedtobediscontinuous.
Hashin’s imperfect interface model [12] assumes that the displacement discontinuities at the flbre/matrix
interface are proportional to the associated interfacial stress; thus the radial, u, and axial, w, displacement
discontinuities are given by
r ¿ (r ) r (cid:190) (r )
w (r )¡w (r )= f rz f and u (r )¡u (r )= f rr f (7)
m f f f m f f f
D D
s n
where subscripts f and m denote flbre and matrix, ¿ (r ) and (cid:190) (r ) are the interfacial shear and normal
rz f rr f
stresses, and D and D are interface parameters for displacement discontinuities in the axial and radial
s n
directions. Notethatwehaveincludedafactorofr ontherighthandsideoftheimperfectinterfacerelations
f
to give D and D units of stress; thus the units of D and D difier from the units used by Hashin [12].
s n s n
As D ! 1 and D ! 1 the interface becomes a perfect interface (zero displacement discontinuities); as
s n
D !0 and D !0, the interface becomes debonded (zero interfacial stress); all other values for D or D
s n s n
are for an imperfect interface.
In Ref. [9], it was argued that the radial stresses in the fragmentation test are predominantly compres-
sive. To prevent the matrix penetrating into the flbre (a negative radial displacement discontinuity), it is
appropriate to set D = 1. In other words, the role of the interface in fragmentation specimens can be
n
modeled with a single interface parameter D . Mathematically, D corresponds to the magnitude of the
s s
axial displacement discontinuity at a two-dimensional interface. Physically it can correspond to any efiect
that in(cid:176)uences flbre/matrix stress transfer. For example there may be an interphase of flnite dimensions
having difierent properties than the bulk matrix, there may be interfacial yielding, there may be debonding
with friction, or there may be cracking into the matrix or flbre. Whatever interfacial mechanism in(cid:176)uences
stress transfer, we argue the results can be flt with an efiective value for D that measures the ability of the
s
interfacial zone or interphase to transfer stress from the matrix to the flbre. Furthermore, we argue that D
s
is a useful interfacial property. It can, for example, be input into variational mechanics models to predict
382 Alkis Paipetis, Costas Galiotis, Yung Ching Liu, and John A. Nairn
Table 1. Thermal and mechanical properties used for the MEBS and MUS carbon
flbres and for the MY750 epoxy matrix in the calculations in this paper.
Property MEBS and MUS MY750
Diameter (2r ) („m) 6.6
f
Tensile Modulus (E or E ) (GPa) 390 2.6
A m
Transverse Modulus (E ) (GPa) 14
T
Axial Shear Modulus (G or G ) (GPa) 20 0.97
A m
Axial Poisson’s Ratio (” or ” ) 0.20 0.34
A m
Transverse Poisson’s Ratio (” ) 0.25
T
Axial CTE (fi or fi ) (10¡6/–C) -0.36 40
A m
Transverse CTE (fi ) (10¡6/–C) 18
T
Temperature Difierence (¢T) (–C) -100
A B

) )
s s
s s
e e y (1+Æs (p) (l,D(i))æ )
r r ¥ zz,f i s
St y (1+Æs (p) (l,D )æ ) St
e ¥ zz,f s e
br br y (1+(1+f )Æs (p) (l,D(d))æ )
Fi Fi ¥ c zz,f s
( (
l l
æz,f æz,f d d
z z
Æs 0 Æs 0
-f y l
c ¥ i
Distance Distance
Figure1. A.Typicalcalculationoftheaverageaxialstressintheflbreinasingleflbrefragmentfromafragmentationtest
using a constant value of the interface parameter, D . B. A modifled analysis constructed to account for damage zones
s
near flbre breaks and residual compressive stress on the ends of the flbre. D =D(d) in the damage zones near the flbre
s s
breaks and D =D(i) in the central, undamaged zone.
s s
the efiect of the interface on real laminates [12]. Thus an interface parameter measured in a single-flbre
fragmentation test can be used to predict the properties of real laminates.
Figure1Ashowsatypicalcalculationforaxialflbrestressinasingleflbrefragmentfromafragmentation
specimen when D is equal to a constant. Real experimental results show two efiects that cannot be flt
s
to such curves. First, flbre breaks in fragmentation specimens inevitably cause an interfacial damage zone
near the flbre breaks. The stress transfer is difierent in the damage zone than in the undamaged, central
region of the flbre fragment. No single value of D can flt a change in stress transfer rate within a single
s
fragment; thus we flt experimental results using a two-zone model [9]. The two-zone model is illustrated in
Fig. 1B. In brief, we assumeD =D(d) in the damage zone andD =D(i) in the undamaged or intact zone.
s s s s
The two-zone analysis is constructed by analyzing a fragment of length l with D = D(d) and a fragment
s s
of length l with D = D(i). The value of l is selected such that the average flbre stress in the two zones
i s s i
are equal at the end of the observed damage zone of length l . Finally, the calculated stress comes from the
d
D =D(d) results in the damage zones and from the l=l and D =D(i) results in the undamaged zone.
s s i s s
Second, as a result of specimen preparation and resin curing, the flbre initially has non-zero axial stress.
Themagnitudeoftheinitialaxialstresscanvarydependingontheamountofflbreprestretchinginthemould,
thetemperatureofmixingofresinwithhardenerand,flnally,thecuringandpost-curingtemperatures. When
the flbre fractures, there should be a controlled recoil of the flbre. The dynamic flbre recoil depends on the
Stress Transfer Fragmentation Test: Raman Experiments and Analytical Modeling 383
4
MEBS
3
)
m
m
1/ MUS
(
y
sit 2
n
e
D
k
a
e
r
B 1
0
0 1 2 3 4 5
Applied Strain (%)
Figure 2. The flbre break density as a function of applied strain during experiments on MEBS/MY750 and MUS/MY750
fragmentation specimens
magnitude of the applied tensile stress, as well as on the residual thermal fleld and prestretching efiects
which govern the energy dissipation mechanism and therefore the value of residual axial stress. The amount
of recovery of the flbre depends on the strength of the interface. Although the resulting surface of the flbre
breakshouldstillbestress-free,thespatialresolutionoftheLRStechnique(1„m)appearstobeinsu–cient
forthedetectionofthepointofzerostressatthetipoftheflbre. TheBessel-Fourierseriessolutionaccounts
forresidualthermalstresses,butnotforprestretchingefiects. Wethusfltexperimentalresultswithnon-zero
stresses at flbre breaks empirically by superposing an additional compressive stress to make the axial flbre
stress have the form (cid:181) ¶
F(z)
h(cid:190)zz;fi=ˆ1 1¡(1+fc) (8)
F(l=2)
where f the magnitude of the compressive stress observed at flbre breaks as a fraction of the far-fleld flbre
c
stress(i.e.,thestressontheflbrebreakis¡fcˆ1). Inflttingexperimentalresults,thevalueoffc waschosen
to match experimental observations. Finally, as illustrated in Fig. 1, the correction for non-zero flbre-break
stress was used only for the stress in the damage zones. The stress in the central, intact zone was calculated
using the uncorrected stress in Eq. (1).
RESULTS
Fibre Axial Stress Distributions
MEBS/MY750 (SIZED SYSTEM)
Ramanspectrawereobtainedatlargeintervals(200„m)priortoflbrefracturetoassessthestresstransfer
e–ciency over the whole length of the embedded flbre. A residual tensile axial stress of about 0.2 GPa
was measured at the onset of the experiment, which was attributed to flbre prestretching during specimen
preparation. Figure 2 shows the average break density as a function of applied strain. The flrst flbre breaks
appeared at about 0.7% applied strain. The flrst complete flbre fragments within the selected \window"
for Raman measurements were detected at about 1.3% applied strain. Once complete flbre fragments were
detected, accurate stress mappings with a resolution of 1 „m on either side of the flbre fracture point were
undertaken at intervals of about 0.1% applied strain. Some measured stress proflles over multiple fragments
at 1.5%, 2.1%, and 3.05% applied strain are given in Figs. 3{5.
384 Alkis Paipetis, Costas Galiotis, Yung Ching Liu, and John A. Nairn
Figure 3. The axial stress in the flbre along several flbre fragments at 1.5% applied strain during a fragmentation test on
an MEBS/MY750 specimen. The symbols are experimental points. The smooth curves are flts to the experiments using
a two-zone model and the Bessel-Fourier series stress analysis with D(i) =300 MPa and D(d) and l as explained in the
s s d
text and plotted in Fig. 9.
Figure 4. The axial stress in the flbre along several flbre fragments at 2.1% applied strain during a fragmentation test on
an MEBS/MY750 specimen. The symbols are experimental points. The smooth curves are flts to the experiments using
a two-zone model and the Bessel-Fourier series stress analysis with D(i) =300 MPa and D(d) and l as explained in the
s s d
text and plotted in Fig. 9.
Stress Transfer Fragmentation Test: Raman Experiments and Analytical Modeling 385
3.5
MEBS/MY750
3.0
= 3.05%
2.5
a) 2.0
P
G
(
s 1.5
s
e
r
St 1.0
0.5
0.0
-0.5
25.0 25.2 25.4 25.6 25.8 26.0 26.2 26.4 26.6 26.8
Distance (mm)
Figure5. Theaxialstressintheflbrealongseveralflbrefragmentsat3.05%appliedstrainduringafragmentationteston
an MEBS/MY750 specimen. The symbols are experimental points. The smooth curves are flts to the experiments using
a two-zone model and the Bessel-Fourier series stress analysis with D(i) =300 MPa and D(d) and l as explained in the
s s d
text and plotted in Fig. 9.
Atanappliedstrainof1.5%,fourdistinctflbrefracturesandthreecompleteflbrefragmentswereobserved
intheRaman\window"(seeFig.3). Therewasonlyasmallamountofcompressivestressattheflbrebreaks
and there was a hint of interfacial damage zones near flbre breaks. The damage zones are indicated by a
slight change in the rate of stress transfer at distances of about 30 to 40 „m from the flbre breaks (or 6 to 7
flbre diameters). At an applied strain of 2.1% (see Fig. 4) there was an additional flbre break and therefore
an additional flbre fragment. Both the compressive axial stress at the fracture point and the size of the
damagezonesincreasedastheappliedstrainincreasedfrom1.5%to2.1%. Theseresultsindicatesomeform
ofinterfacialfailure[4]hadpropagatedfurtherastheappliedstrainincreased. By3.05%(seeFig.5)applied
strain, the damage zones became very pronounced and extended distances of about 100 „m from each flbre
break. The damage zones became even more apparent at applied strains higher than 3.05%.
MUS/MY750 (UNSIZED SYSTEM)
As done with the MEBS sized system, Raman spectra for the MUS unsized system were obtained at
large intervals (200 „m) prior to flbre fracture to assess the integrity and stress transfer e–ciency of the
whole coupon. A residual compressive axial stress of about ¡0:3 GPa was measured at the onset of the
experiment. This stress was attributed to thermal shrinkage of the resin during curing, which was not ofiset
by flbre prestretching during specimen preparation (as was the case for the MEBS system). Figure 2 shows
the average break density as a function of applied strain. The flrst flbre breaks appeared at about 0.6%
applied strain. The saturation break density was lower (or the critical fragment length was longer) for the
MUS system then for the MEBS system. The flrst complete flbre fragments within the selected \window"
for Raman measurements were detected at about 1.2% applied strain. Once complete flbre fragments were
detected, accurate stress mappings with a resolution of 1 „m on either side of the flbre fracture point were
undertaken at intervals of about 0.1% applied strain. Some measured stress proflles over multiple fragments
at 1.6%, 2.1%, and 2.65% applied strain are given in Figs. 6{8.
At an applied strain of 1.6%, six distinct flbre fractures and flve complete flbre fragments were observed
in the Raman \window" (see Fig. 6). The residual compressive stress at the flbre breaks was larger for
MUS specimens than at the corresponding strain with MEBS specimens. This larger compressive stress
may have been caused by the difiering initial axial stresses prior to the experiment. At 1.6% applied strain
there were indications a interfacial damage zones near flbre breaks of about 40 „m (7 or 8 flbre diameters).
386 Alkis Paipetis, Costas Galiotis, Yung Ching Liu, and John A. Nairn
3.5
MUS/MY750
3.0
= 1.6%
2.5
2.0
)
a
P
G 1.5
(
s
es 1.0
r
St
0.5
0.0
-0.5
-1.0
19.2 19.6 20.0 20.4 20.8 21.2 21.6
Distance (mm)
Figure 6. The axial stress in the flbre along several flbre fragments at 1.6% applied strain during a fragmentation test on
an MUS/MY750 specimen. The symbols are experimental points. The smooth curves are flts to the experiments using
a two-zone model and the Bessel-Fourier series stress analysis with D(i) =150 MPa and D(d) and l as explained in the
s s d
text and plotted in Fig. 9.
At an applied strain of 2.1% (see Fig. 7) there were an identical number of flbre breaks and fragments in
the same Raman \window". As with the MEBS system, however, all damage zones increased in size as the
applied strain increased from 1.6% to 2.1%. Finally, at 2.65% applied strain, the damage zones became very
pronounced and extended distances of about 90 „m from each flbre break.
Two-Zone Bessel-Fourier Data Analysis
All experimental Raman results were analyzed by fltting to the two-zone model in Fig. 1B. For both
MEBS and MUS, we sought flts for which stress transfer in the central, intact zone was independent of the
damage that had occurred near the flbre breaks. In other words, D(i) was taken as a strain-independent
s
material property of the interphases in each specimen type. Furthermore f was set to zero for the central,
c
intact zones. The intact zones of all MEBS specimens could be flt using D(i) = 300 MPa; the intact zones
s
of all MUS specimens could be flt using D(i) = 150 MPa. We emphasize that although D(i) was assumed
s s
to be constant in the fltting process, the fact the we could flt all results successfully means that a constant
D(i) is an experimental flnding. In other words, we could not resolve changes in D(i) for a single flbre type
s s
byallowingittobeastrain-dependentquantity. ThedifierencesbetweentheD(i) valuesforthesetwoflbres
s
are signiflcant in that we have resolved difierences in stress transfer properties; the results show that stress
transferismoree–cientinthesizedMEBSspecimenthanintheunsizedMUSspecimens. Thefactoroftwo
difierence, however, may overstate the difierences. The change in stress transfer rate is not linearly related
to D(i). Some example plots in Ref. [9] show the efiect of D(i) on stress transfer rate over the range of
s s
D(i) from 0 to 1. the difierence between a D(i) = 300 MPa and a D(i) = 150 MPa are resolvable, but not
s s s
dramatic.
When including the damage zones in the analyses, there were three factors to consider. First, we needed
to select the length of the damage zone, l . It is di–cult to determine l directly from Raman data or
d d
by microscopy observations using photoelasticity. We argue, however, that our fltting process made the
selection of l less ambiguous. If l was chosen incorrectly, the flts in the central, intact zone would sufier;
d d
by monitoring the entire fltting process we were thus able to narrow in on the correct l or at least a value
d
of l that is consistent with all other experimental observations. Second, there was often some asymmetry
d